Oberseminar Nonlinear Dynamics

Organizers

Program

April 16, 2013

Pavel Gurevich(Free University Berlin)

Hysteresis with diffusive thresholds

We describe a system with fluctuating hysteresis thresholds by means of diffusion equations, where the role of the spatial variable is played by hysteresis threshold.
To illustrate our approach, we consider a prototype model that describes a population of two-phenotype individuals in a varying environment. The key features are as follows:

The individuals can switch between the two phenotypes by hysteresis law (non-ideal relay).

The thresholds for each individual can fluctuate. Under the assumption that this fluctuation obeys the Gaussian distribution, we arrive at a system where the density of the population obeys a reaction-diffusion equation with discontinuous hysteresis.

The collective impact of the population on the environment is described in terms of the Preisach operator with a measure given by the population density. This measure becomes time dependent and is now a part of the solution.

In the talk, we formulate a well-posedness theorem and discuss emerging spatial patterns.

April 23, 2013

Serhiy Yanchuk(Humboldt University of Berlin)

Delay induced patterns in 2D neuronal lattices

April 30, 2013

Marek Fila(Comenius University Bratislava)

Extinction of solutions of the fast diffusion equation

We consider positive solutions of the Cauchy problem for the fast diffusion equation. Sufficient conditions for extinction of solutions in finite time are well known. We shall discuss results on the asymptotic behavior of solutions near the extinction time obtained in collaboration with John R. King, Juan Luis Vazquez, Michael Winkler and Eiji Yanagida. We shall pay particular attention to a critical case with slow asymptotics.

Darbo established ub 1955 a fixed point theorem in which the
compactness hypothesis of Schauder's fixed point theorem was
dramatically relaxed. Essential Maps, introduced by A. Granas
and also by M. Furi, M. Martelli, and A. Vignoli, are an extension
and homotopic analogue of degree theory. Both ideas were married,
historically with very technical proofs, for applications in
so-called nonlinear spectral theory.
In the talk a stunningly simple combination of the two ideas
is presented. This even applies to a new concept of so-called
essential pairs which appears to be the natural extension
and homotopic analogue of recently developing degree theories
for function triples.

May 28, 2013

Haibo Ruan(University of Hamburg)

A degree theory for synchrony-breaking bifurcations in coupled cell systems

A network is a graphical entity consisting of nodes and links between the nodes.
In the theory of coupled cell systems, nodes of the network are interpreted
as individual dynamical systems whose mutual interactions are described
by the coupling structure of the network.
One important and most studied collective dynamics on networks is the synchronization:
a set of cells is said to be synchronized, if their individual dynamics coincide over time.
Fully synchronized states where all cells are in synchrony, are rare instances.
The more common phenomenon is partial synchronization where communities or clusters of cells are synchronized.
A synchrony-breaking bifurcation refers to a local bifurcation,
where a fully synchronous equilibrium loses its stability and bifurcates to states of less synchrony.
In this talk, we introduce a lattice degree which can be used to study synchrony-breaking bifurcations,
both of steady states and of oscillating states, and to predict the existence of these bifurcating branches,
together with their multiplicity, symmetry and synchrony types.

It is known that the Swift-Hohenberg equation, which is a fourth-order PDE, can be reduced to the Ginzburg-Landau equation (amplitude equation), which is a second-order PDE, by means of a singular perturbation method. In this talk, a reduction of a certain class of (a system of ) nonlinear parabolic equations is proposed based on the renormalization group method, which is one of the perturbation methods applicable to a wide class of ODEs/PDEs.

June 25, 2013

Atsushi Mochizuki(RIKEN, Tokyo)

Phenomenology of rate sensitivity in reaction networks

One experimentally possible procedure to understand the dynamics of
chemical reaction systems (including metabolic networks) is based on
perturbations of reaction rates. For example we may increase/decrease or knockout
enzymes that mediate reactions in the systems. Frequently, however, the experimental response to
perturbations seems difficult to understand.
In this talk, we present a mathematical approach to determine
the equilibrium response from the structure of the network.
In particular, some perturbations may cause large deviations,
while other perturbations may go unnoticed.